Asymptotics of counts of small components in random structures and models of coagulation-fragmentation

Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
mar18aa@techunix.technion.ac.il

Received: 4 November 2010
Revised: 18 September 2011

Abstract

We establish necessary and sufficient conditions for the convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. The multiplicative measures depict distributions of component spectra of random structures and also the equilibria of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We show that the convergence of multiplicative measures is equivalent to the asymptotic independence of counts of components of fixed sizes in random structures. We then apply Schur’s tauberian lemma and some results from additive number theory and enumerative combinatorics in order to derive plausible sufficient conditions of convergence. Our results demonstrate that the common belief, that counts of components of fixed sizes in random structures become independent as the number of particles goes to infinity, is not true in general.